## Eccentric Lagrange Point Simulations

If the Java applet fails to start due to *Java Security* issues,
click
here.

It is known that for the planar restricted (i.e., one body has no mass)
3-body problem, the L4/L5 Lagrange
points are stable when (and only when) m0 is at least 24.96 times as massive as
m1 (or the other way around). However, these orbits exist for any mass ratios
and any eccentricity. Whether they are stable is another matter. This appliet
allows the user to investigate which scenarios are stable and which are not.
The mathematics describing how to find the initial positions and velocities is
given in
Lagrange Points for Eccentric Planar 3-Body Systems.

The stability analysis for the restricted circular case can be found in almost
any textbook on celestial mechanics. You can also read about it here:
Linear Stability of Lagrange Points: Complex Variable Notation.

In addition to giving the usual result (with a perhaps unusual and definitely
streamline development), I also consider the stability of L4/L5 in a "classical
flatland". That is, in a universe where the force of gravity decays like one
over distance rather than one over distance squared. In flatland, L4 and L5
are always stable. There is no condition on the ratio of the masses.